Chapter 11: The Radiative Transfer Equation with Scattering Key Points: Single and multiple scattering definitions. Scatter once, single, scatter several times, multiple. When does single scattering matter? Always. It may be small relative to absorption, but generally should consider it. When is multiple scattering important? When the optical depth for scattering is ‘large enough’, greater than say 0.5. General form of the multiple scattering equation. Single scattering approximation. Review of the phase function, and asymmetry parameter.

Demonstrations Diffraction grating from a DVD. Grazing incidence, first order (over head) and second order (near backscattering). Milk, Clouds, and multiple scattering: Rayleigh scattering by dilute milk, and polarization state.

Review: Optics of N identical (particles / volume) Light beam area = A z dz z+dz Power removed in dz: = I(z) N A dz ext Bouger-Beer “law” (direct beam only!)

CH 8: ATMOSPHERIC EMISSION: PRACTICAL CONSEQUENCES OF THE SCHWARZSCHILD EQUATION FOR RADIATION TRANSFER WHEN SCATTERING IS NEGLIGIBLE What process subtracts radiation? What process adds radiation? What equation is used to calculate optical depth for a gaseous atmosphere?

Visibility Overview: From Bill Malm, NPS Fort Collins Scattering of object rays from the sight path, and scattering of stray light into the sight path, both by gases and aerosols. Important factors involvede in seeing a scenic vista. Image forming information from an object is reduced (scattered and absorbed) as it passes through the atmosphere to the human observor. Sunlight, light from clouds and the grounds are scattered from particles in the sight path. Some scattered light remains in the sight path and can be so bright that the image essentially disappears. Visibility Overview: From Bill Malm, NPS Fort Collins

Putting it all together... General Radiative Transfer Equation ds I(s,) I(s+ds,) I(s,') I(s,’’) Phase function describes the fraction of light that is scattered into  from other directions. gains losses lost only from the forward direction !! new term

General Equation for Radiative Transfer

Multiple Scattering Complex Due to Multiple Scatterings Hypothetical Photon Path

Single Scattering Approximation I0 From Sun R1 phase function scattering coefficient  R3 R2

Rayleigh Scattering sunset

Angular Dependence of Rayleigh Scattering (dipole) From: http://qels.com/theory/rayleighscattering/mass.cfm http://www.bo.astro.it/sait/spigolature/spigo402base.html Horizontal E-field Vertical E-field Dipoles don’t radiate in the direction they are undergoing linear ocsillation. From http://www.sparknotes.com/physics/optics/phenom/section3.rhtml

Rayleigh and Raman Scattering: Quantum Perspective One in a million photons are Raman scattered. Rayleigh scattering merely disperses radiation in space. Stokes heats, Anti Stokes cools.

Rayleigh Scattering (light scattering by air as dipole radiation) From Liou pg 93.  is the molecular anisotropy parameter as the polarizability is really a tensor. The refractive index relationship is in relation to the polarizability of air.  = 0.035 for air. Dry air, 15 C, 101325 Pa, 0.045% CO2 by volume, vacuum  in microns, (Birch, Metrologia, 1994, 31, 315). From http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html. Dry air, t in Celcius, P in Pascal, 0.045% CO2 by volume, Birch, Metrologia, 1994, 31, 315). From http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html. Number concentration of air molecules.

Dry Air Refractive Index

Rayleigh Scattering (light scattering by air as dipole radiation)

Rayleigh Scattering (light scattering by air as dipole radiation) 42.3% of Total Energy, TOA

Rayleigh Scattering (light scattering by air as dipole radiation) 1.5% of Total Energy, TOA

Rayleigh Scattering (light scattering by air as dipole radiation) 6.3% of Total Energy, TOA

Rayleigh Scattering (light scattering by air as dipole radiation) 0.5% of Total Energy, TOA

Rayleigh Scattering In Perspective Relative to Absorption

Rayleigh Scattering Intensity as a function of Scattering Angle. q I0 Isca(q) N scatterers / volume Random E-field incident, random scatterer orientation. From: http://www.jenkinsdisplays.com/led_bulbs/rayleigh_scattering.html

Dipole Radiation Pattern: (Petty, Ch12). Incident E-field vertical: Dipole charge oscillation vertical.  = incident direction Irradiance Average for Random E-field: sum of the polarized patterns / 2. Incident E-field Horizontal

Aside: Asymmetry Parameter of Scattering, g. -1<g<1 nr=1.33 =0.6328 D=20 um g=0.874 Is()  I0

‘Typical’ Water Droplet Cloud Optical Properties Deff = 20 um Variance = 0.1 Why does the single scatter albedo go so low at around 3 microns? Why does the asymmetry parameter go so large at around 3 microns? COMPLEX REFRACTIVE INDEX OF WATER: Visible in black.

Optics of Visibility: I=Radiance, Radiant Intensity C= Visual Contrast = (Isurroundings- Iobject ) / Isurroundings object Surroundings from http://vista.cira.colostate.edu/improve/Education/Workshops/WESTAR/Malm/partiii_malm.ppt

The Anatomy of the Human Visual System The human eye Eye diameter is about 25mm Cornea protects the eye from the outside and provides most of the optical focusing power Aqueous humor provides nourishment to cornea and lens Iris regulates the amount of light that is allowed into the eye Pupil is the aperture left open by the iris Small pupil under high luminance conditions provides sharp image (pinhole camera effect) Adjustment of pupil gets more difficult with age Focal length of lens is adjusted by ciliary muscle focal distance, diopters (1/focal length) lens gets harder with age and accommodation becomes more difficult from http://arrow.win.ecn.uiowa.edu/56147/LECTURE/c-20.ppt

Daytime and Nightime Sensitivity of the Eye From Proctor and VanZandt (1993) Daytime and Nightime Sensitivity of the Eye from http://arrow.win.ecn.uiowa.edu/56147/LECTURE/c-20.ppt

Instrument for Visibility Measurement (not the bird) ... from http://vista.cira.colostate.edu/improve/Education/Workshops/WESTAR/Malm/partiii_malm.ppt

Calculation of Contrast: Horizontal Light Path Through a Hazy Atmosphere Key: J is common to both terms in the numerator in C. I’(0) J s dirty atmosphere I(0) observer object light shines on the object, background, and atmosphere How far, s, can observer see???? surroundings C= Visual Contrast = (Isurroundings- Iobject ) / Isurroundings C = ( I’(s) - I(s) ) / I’(s) = exp(-exts) ( I’(0) - I(0) ) / [exp(-exts) I’(0) + J(1- exp(-exts))]

Calculation of Contrast: Horizontal Light Path Through a Hazy Atmosphere Key: J is common to both terms in the numerator in C. I’(0) J s dirty atmosphere I(0) = 0 observer black object light shines on the object, background, and atmosphere How far, s, can observer see???? white surroundings C= Visual Contrast = (Isurroundings- Iobject ) / Isurroundings C = ( I’(s) - I(s) ) / I’(s) = exp(-exts) I’(0) / [exp(-exts) I’(0) + J(1- exp(-exts))]

Calculation of Contrast: Horizontal Light Path Through a Hazy Atmosphere Key: J is common to both terms in the numerator in C. F0 I’(0) J s dirty atmosphere I(0) = 0 observer black object light shines on the object, background, and atmosphere How far, s, can observer see???? white surroundings I’(0)≈ F0

Calculation of Contrast: Horizontal Light Path Through a Hazy Atmosphere Key: J is common to both terms in the numerator in C. I’(0) J s dirty atmosphere I(0) = 0 observer black object light shines on the object, background, and atmosphere How far, s, can observer see???? white surroundings C = ( I’(s) - I(s) ) / I’(s) = exp(-exts) I’(0) / [exp(-exts) I’(0) + J(1- exp(-exts))] C ≈ 0.02, > 0.02 for people

Calculation of Contrast: Horizontal Light Path Through a Hazy Atmosphere Key: J is common to both terms in the numerator in C. I’(0) J s dirty atmosphere I(0) = 0 observer black object light shines on the object, background, and atmosphere How far, s, can observer see???? white surroundings What makes s small????